Base or Radix Converter

A base converter, also called a radix converter, is a tool that rewrites a number in a different numeral system without changing the value it represents. Base and radix refer to the same idea: the count of unique digits a number system uses to write its values.

This guide covers the number systems people run into most often, binary, decimal, octal, and hexadecimal, along with extended and named bases such as duodecimal and sexagesimal, and touches on non-integer and negative bases as well. It works through the methods used to move between these systems, positional value, Horner's method, repeated division, direct grouping between related bases, fractional numbers, negative numbers, and mixed-radix systems, and flags the errors that most often trip people up during a manual conversion.

What Are the Main Number Systems Used in Base Conversion?

A base, or radix, is the number of unique digits a number system uses before it rolls over into a new position. Binary has a base of 2, decimal has a base of 10, and so on. Every digit in a number occupies a position, and that position carries a value equal to the base raised to a power, an idea usually called positional value or place value.

Base 10 feels natural because humans have ten fingers to count on, so counting in tens became the default long before anyone thought about number theory. Computers use base 2 instead because a transistor or switch only has two reliable states, on and off, which map cleanly onto the digits 0 and 1.

Are "Base" and "Radix" the Same Thing?

Yes, base and radix mean the same thing. Radix is the more formal, mathematical term, while base is the word most people reach for in everyday conversation, and both describe the number of unique digits a number system uses. Different tools and articles switch between the two words without explaining why, which is usually where the confusion comes from.

What Digits Does the Binary Number System (Base 2) Use?

Binary uses only two digits, 0 and 1, and each position in a binary number represents a power of 2. It is the native language of computer circuits because a transistor or switch has two clean, unambiguous states, on and off, which map directly onto those two digits.

Reading a binary number is a matter of adding up the powers of 2 that are switched on. Take 1011: reading right to left, the digits sit in the 1s, 2s, 4s, and 8s positions, so the number equals 8 + 0 + 2 + 1, or 11 in decimal.

Binary numbers grow long fast even for modest values, which is exactly why octal and hexadecimal exist, as shorter, more readable stand-ins for the same binary data.

The row above maps binary 00001011 across the standard 8-bit place values, which reduces to decimal 11.

What Digits Does the Decimal Number System (Base 10) Use?

Decimal uses ten digits, 0 through 9, and it is the system used for everyday counting and arithmetic. It is also the base most conversions start from or land on, since it is the reference point everyone already understands intuitively. As with any positional system, each digit in a decimal number is multiplied by a power of 10 based on where it sits.

What Digits Does the Octal Number System (Base 8) Use?

Octal uses eight digits, 0 through 7, and each octal digit corresponds to exactly three binary digits. It was more common in early computing and today survives mainly in Unix and Linux file permission codes.

A short example: the octal number 17 equals (1 x 8) + 7, or 15 in decimal.

What Digits Does the Hexadecimal Number System (Base 16) Use?

Hexadecimal uses sixteen symbols, the digits 0 through 9 plus the letters A through F, where A equals 10 and F equals 15. Each hexadecimal digit maps to exactly four binary digits, which is why a single byte, 8 bits, always converts to exactly two hex digits.

This compactness is the main reason hex shows up so often in programming: it is a shorter, more readable stand-in for long strings of binary. As a quick example, the hex number 2F equals (2 x 16) + 15, or 47 in decimal.

How Do Custom Bases Beyond Base 16 Represent Digits?

Number systems are not limited to base 2, 8, 10, and 16. Bases above 10 continue the same pattern by using letters after 9: uppercase A through Z extends the digit set up to base 36, and lowercase a through z can extend it further, up to base 62. From base 37 upward, uppercase and lowercase letters count as different digits, since the digit set becomes case sensitive at that point.

These custom bases show up in practice more than people expect. Base 36 and base 62 encoding are both used to shorten identifiers, most visibly in short URL codes. In base 36, the number 35 is represented by the single digit Z; base 62 adds the lowercase letters on top of that to squeeze even more values into fewer characters.

What Are Some Named Number Bases Used in Real Life, Like Base 12, Base 20, and Base 60?

Some number bases carry their own names because they map onto something familiar outside of computing.

Can a Number Base Be Non-Integer or Negative?

Every example so far uses a whole number base, but some non-standard systems use a non-integer base, such as base 1.5 or the golden ratio base, or a negative base, such as base -2. These are rare and mostly confined to mathematics and computer science theory rather than everyday use, but the underlying idea does not change: each digit position still represents a power of the base, it is just that the base itself is no longer a positive whole number.

How Do You Convert a Number From One Base to Another?

There are two main directions for converting between bases: converting into decimal from another base, and converting out of decimal into another base. There is also a shortcut for moving directly between binary, octal, and hexadecimal without passing through decimal at all. Each method is covered below, with a worked example to follow along with.

How Do You Convert Any Base to Decimal Using Positional Value?

The positional value method works the same way regardless of which base you start from:

1. Write out each digit of the number along with its position, counting from zero on the right.

2. Multiply each digit by the base raised to the power of its position.

3. Add all the results together.

Worked example: converting the hexadecimal number C9 to decimal.

192 + 9 = 201, so hexadecimal C9 equals decimal 201. The same steps apply to binary, and a binary to decimal converter runs through them automatically for any length of binary input.

Is There a Faster Way to Convert Any Base to Decimal Using Horner's Method?

Horner's method, sometimes called the accumulation method, reaches the same answer with fewer separate multiplications, which is why it is faster to do by hand and is what most calculators and programs use internally.

1. Start with a running total of zero.

2. Scan the digits from left to right.

3. At each digit, multiply the running total by the base, then add the current digit.

Worked example: converting the base 7 number 245 to decimal.

Base 7 245 equals decimal 131.

How Do You Convert Decimal to Any Other Base Using Repeated Division?

1. Divide the decimal number by the target base.

2. Record the remainder.

3. Repeat the division on the quotient until it reaches zero.

4. Read the remainders from bottom to top for the final answer.

Worked example: converting decimal 13 to binary.

Reading the remainders from bottom to top gives 1101, so decimal 13 equals binary 1101. If any step produces a digit that does not exist in the target base, a mistake happened somewhere earlier in the division. The same repeated division process is what sits behind a decimal to hexadecimal converter, just with 16 as the divisor instead of 2.

Is There a Manual Shortcut for Converting Between Decimal and Octal?

Converting between decimal and octal by hand has a shortcut that avoids long division, based on the fact that ten equals eight plus two. Powers of ten can be rewritten in terms of powers of eight, which makes the arithmetic noticeably easier to manage on paper or in your head compared to plain repeated division. It is worth knowing if you convert between the two often, but repeated division works fine as the default method.

How Do You Convert Directly Between Two Bases Without Going Through Decimal?

When one base is a power of another, base 16 is a power of base 2, and base 9 is a power of base 3, you can convert directly between them without going through decimal at all, by grouping digits in the smaller base into sets that match the power relationship.

The general rule: to convert from base b to base b raised to the power k, group the digits in sets of k starting from the right, then convert each group individually. Binary to hex and binary to octal, covered next, are both specific applications of this same rule. The same principle applies to other pairs too, such as base 3 to base 9.

How Do You Convert Binary to Hexadecimal Using Bit Grouping?

Split the binary number into groups of four digits, starting from the right and padding with zeros on the left if the total digit count does not divide evenly. Convert each group directly to its hex digit.

Worked example: converting binary 10011100 to hex.

Binary 10011100 equals hex 9C. This works because each hex digit always equals exactly four binary digits. The reverse direction, hex to binary, is the same grouping process run backward: expand each hex digit into its four-bit binary equivalent. A binary to hexadecimal converter handles both directions instantly for longer binary strings.

How Do You Convert Binary to Octal Using Bit Grouping?

The same idea applies for octal, except the grouping happens in sets of three digits from the right, again padding with zeros if needed. Convert each group to its octal digit.

Worked example: converting binary 101110011 to octal.

Binary 101110011 equals octal 563. This works because each octal digit always equals exactly three binary digits. For longer binary strings, a binary to octal converter groups and converts the digits automatically.

How Do You Convert a Fractional Number to Another Base?

Numbers with a decimal point are converted in two separate parts, the whole number part and the fractional part, using different methods for each. For the whole number part, use repeated division as covered earlier. For the fractional part, use repeated multiplication: multiply the fraction by the target base, record the whole number digit produced, then repeat on the remaining fractional part.

Worked example: converting decimal 0.625 to binary.

Decimal 0.625 equals binary 0.101. Some fractions never terminate this way and instead repeat forever, and rounding too early in the process is one of the most common sources of wrong answers.

How Do You Convert Negative Numbers Between Bases?

The simple approach: convert the absolute value of the number normally, then attach the negative sign to the result. This works fine for most everyday conversions.

In computing, negative numbers are more often represented using two's complement rather than a plain sign. Two's complement is produced by flipping every bit of the positive value and then adding one. It is a different context from the sign-based method above, built for how processors handle subtraction, and the two methods are worth keeping separate rather than treating as interchangeable.

What Is a Mixed-Radix Number System and How Is It Different From Standard Base Conversion?

A mixed-radix number system does not use the same base for every digit position, which sets it apart from everything covered so far. Time is the clearest everyday example: seconds and minutes each roll over at 60, but hours roll over at 24, so a duration like 1 day, 2 hours, 15 minutes, and 30 seconds is really a mixed-radix number built from the bases 24, 60, and 60.

The same repeated division idea from standard base conversion still applies here, except the base changes at every step instead of staying fixed.

What Common Errors Happen When Converting Between Number Bases?

Most base conversion mistakes fall into a small number of repeatable categories. Recognizing the pattern is often enough to catch the error and fix it.

What Does an Illegal Digit Error Mean in Base Conversion?

An illegal digit error happens when a result contains a digit that does not exist in the target base, an 8 or 9 showing up in a base 7 or base 8 answer, for instance. It is one of the fastest ways to catch a mistake, since a digit outside the valid range for that base signals immediately that something went wrong during division or grouping. A base 8 result reading 289 is impossible on its face, since 8 and 9 do not exist as octal digits.

How Can You Check a Base Conversion Result Is Correct?

Two habits catch most errors before they cause real problems.

The digit count rule: converting to a lower base can never produce fewer digits than the original, and converting to a higher base can never produce more digits. A result that breaks this pattern is a signal something went wrong.

The reverse-conversion check: convert the answer back to the original base, or to decimal and then back, and confirm it matches the starting number.

Why Do Fractional Conversions Sometimes Repeat Forever Instead of Terminating?

A fractional part fails to terminate when the target base does not share the same underlying factors as the original base, so the digits repeat forever no matter how many steps get calculated. Decimal 0.1 is a simple example: converted to binary, it produces an endlessly repeating pattern rather than a clean stopping point.

This is expected behavior for certain base and fraction combinations, not a mistake, and any conversion method or tool has to round or truncate at some point when it happens.

Why Do Some Converters Fail on Large Numbers?

Many free online converters silently produce wrong results on very large numbers because they rely on standard number types with limited precision instead of arbitrary precision arithmetic. A reliable tool should support large integers explicitly and never round silently in the background.

Where Is Base Conversion Used in Real Computing Tasks?

Base conversion shows up across a wide range of everyday computing tasks. Networking relies on it because IP addresses and subnet masks are really binary numbers written out in decimal or hex for human convenience. System administration leans on octal for Unix file permissions. Web design uses hexadecimal for color values. Text encoding maps characters onto numeric codes that then get converted into whichever base a system needs. Each of these gets a closer look below.

How Do You Convert an IP Address to Binary for Subnetting?

An IPv4 address is really a 32-bit binary number split into four 8-bit sections called octets. The familiar dotted decimal format, 192.168.1.1 for example, exists purely for human readability.

Take one octet, 192, and convert it to binary using repeated division: 192 equals 11000000 in 8-bit binary.

This matters for subnetting because subnet masks work by comparing binary bit patterns rather than decimal numbers. For the fuller networking math behind subnet masks and host ranges, an IP address to binary converter handles the calculation directly.

How Does Octal Represent Unix File Permissions Like chmod 755?

Unix and Linux file permissions are commonly written as a three digit octal number, where each digit represents read, write, and execute permissions for owner, group, and others.

This ties directly back to octal-to-binary conversion covered earlier: each octal digit here is really a 3-bit binary pattern of permission flags. Setting permissions directly is easier with a chmod permissions calculator rather than working the octal out by hand every time.

How Do You Convert a Hex Color Code to Decimal RGB Values?

Web colors written as hex codes, #FF5733 for example, are really three separate hexadecimal numbers, one each for red, green, and blue, each ranging from 00 to FF.

Take the pair FF: using the same positional value method covered earlier, F x 16 + F equals 255 in decimal.

Designers and developers move between hex and decimal RGB values constantly, since adjusting brightness or opacity programmatically usually calls for decimal values rather than hex. A hex to RGB color converter handles this conversion directly for any hex code.

How Do You Convert Text to Binary Using ASCII Values?

Every text character has a numeric code behind it, historically defined by the ASCII standard, and that number can then be converted into binary, octal, or hexadecimal using the exact same methods covered earlier.

Take the letter A: its ASCII decimal value is 65, which converts to binary as 01000001.

This is a two-step process, character to number first through ASCII, then number to another base through standard conversion, which is worth keeping separate in your head since base conversion only covers the second half of the task. A text to binary converter runs both steps automatically for any string of characters.

How Does Hexadecimal Represent Computer Memory Addresses?

Computer memory locations are identified by addresses that are naturally very large binary numbers. Hexadecimal writes them in a much shorter, more readable form, since each hex digit stands in for four binary digits.

A memory address that runs to 32 binary digits collapses to just 8 hexadecimal digits, which is the entire reason hex became the standard shorthand for this kind of value. For deeper technical context on hexadecimal notation in programming, a hexadecimal programming reference covers the terminology in more depth.

How Is Base 64 Different From Base Conversion Used in Encoding?

Despite the name, Base64 encoding is not a true positional number base conversion like the ones covered throughout this guide. It is a scheme for representing binary data as readable text using a set of 64 characters, which is a different problem from rewriting a number's value in a different numeral system.

Base64 shows up most often in embedding images inside text formats or encoding data for transmission where only plain text characters are allowed. A Base64 encoder and decoder handles that encoding and decoding directly.

How Is Converting to Roman Numerals Different From Converting Between Number Bases?

Roman numeral conversion looks similar to base conversion on the surface, since it also changes how a number is written, but it is fundamentally different because Roman numerals are not a positional base system. There is no radix and no place value; each symbol carries a fixed value that gets added or subtracted based on its position relative to neighboring symbols.

Roman numerals still show up regularly for formatting dates, chapter numbers, or clock faces. A Roman numeral converter handles that conversion directly when you need it.